A First Course in Topology

by McCleary, John

• ISBN13: 9780821838846
• ISBN10: 0821838849
• Format: Paperback
• Copyright: 2006-03-01
• Publisher: Amer Mathematical Society
• More Book Details

Summary

How many dimensions does our universe require for a comprehensive physical description? In 1905, Poincare argued philosophically about the necessity of the three familiar dimensions, while recent research is based on 11 dimensions or even 23 dimensions. The notion of dimension itself presented a basic problem to the pioneers of topology. Cantor asked if dimension was a topological feature of Euclidean space. To answer this question, some important topological ideas were introducedby Brouwer, giving shape to a subject whose development dominated the twentieth century. The basic notions in topology are varied and a comprehensive grounding in point-set topology, the definition and use of the fundamental group, and the beginnings of homology theory requires considerable time. Thegoal of this book is a focused introduction through these classical topics, aiming throughout at the classical result of the Invariance of Dimension. This text is based on the author's course given at Vassar College and is intended for advanced undergraduate students. It is suitable for a semester-long course on topology for students who have studied real analysis and linear algebra. It is also a good choice for a capstone course, senior seminar, or independent study.

Table of Contents

A little set theory

Metric and topological spaces

Geometric notions

Building new spaces from old

Connectedness Compactness

Homotopy and the fundamental group

Computations and covering spaces

The Jordan Curve Theorem

Simplicial complexes Homology

Bibliography

Notation Index

Subject Index

Table of Contents provided by Publisher. All Rights Reserved.